First attested in the Seokbo sangjeol (釋譜詳節 / 석보상절), 1447, as Middle Korean봄〮 (Yale: pwóm). The etymology beyond Middle Korean is not clear:
Middle Korean nouns ending in /-m/ are often substantives of verbs, but the only relevant verb is 보다〮 (Yale: pwò-tá, “to see”), which seems semantically unrelated.
Some online sources claim a connection to the element 봄 (Yale: pwom) in the Middle Korean compound verb 봄놀다 (Yale: pwom-nwolta, “to romp; to spring about in play”), whose second element is 놀〯다〮 (Yale: nwǒltá, “to play”). If valid, the semantics would be similar to Englishspring. It is not clear whether this has been suggested in academia.
However, it appears more likely that 봄 (Yale: pwom) is an allomorph of *봅 (Yale: pwop), nasalized by the following /n/; compare ablaut variant 붑괴다 (Yale: pwup-kwoyta, “to seethe”).
It has been tentatively proposed that this is an ancient pre-Sino-Korean borrowing from Old Chinese芃 (OC *[b]ˤ[o]m, “luxuriant growth (of foliage)”).